Mohammad-Javad Beigrezaee, Ph.D. Candidate 37th cycle, University of Trento, DICAM

The current paper extends the classic power law Gibson-Ashby model of cellular solids for repetitive graded lattices. Three well-known porous unit cells, i.e., simple cubic (SC), body-centered cubic (BCC), and their combination (BCC+), are considered and the corresponding graded lattices are geometrically defined by functional variation in the diameters of the ligaments based on a power law. The analytical expressions for the relative elastic modulus of the cells, which vary along the graded direction, are provided by structural analysis of the representative beam framework containing tapered beams using the Euler-Bernoulli theory. After that, continuous approximation functions for variation of the relative density and the relative elastic modulus are presented by curve fitting over the analytical expressions along the graded direction. Then, the graded lattices are considered to be fully homogenized and as a homogeneous media, the exact expressions for the effective relative density and the effective relative modulus are presented. The modified version of the Gibson-Ashby model is proposed to relate the effective relative modulus to the effective relative density by introducing a correction factor to the classic model. The calibrated correction factor is given for a wide range of changes in the diameters of ligaments and the power of graded variations law and for three types of unit cells. It is shown that the classic Gibson-Ashby model with the coefficients fitted on uniform lattices is not accurate for relating the effective properties of the graded lattices. In addition, it is observed that the modified model is able to fit more accurately the local (cell by cell) relative elastic modulus to the local relative density along the graded direction compares to the classic one, especially for the BCC lattice. The proposed model in form of analytical expression accompanied by the fitting parameters and the correction factor provided for a wide range of variations in the geometry of graded lattices can be considered as a guideline for designing optimized functionally graded lattice structures.
Sample graded lattice structure with a linear variation of dimensionless diameter of ligaments R, from R0=0.05 to RN=0.25 for a) SC, b) BCC and c) BCC+ unit cells.